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Theorem icomnfordt 18842
Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
icomnfordt  |-  ( -oo [,) A )  e.  (ordTop `  <_  )

Proof of Theorem icomnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2443 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2443 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 18839 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 18832 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2514 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 18597 . . . . . . 7  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  <-> 
( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top )
86, 7mpbir 209 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 18593 . . . . . 6  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  ->  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
) )
108, 9ax-mp 5 . . . . 5  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
1110, 4sseqtr4i 3410 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3540 . . . . 5  |-  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) 
C_  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
13 ssun2 3541 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2443 . . . . . . . 8  |-  ( -oo [,) A )  =  ( -oo [,) A )
15 oveq2 6120 . . . . . . . . . 10  |-  ( x  =  A  ->  ( -oo [,) x )  =  ( -oo [,) A
) )
1615eqeq2d 2454 . . . . . . . . 9  |-  ( x  =  A  ->  (
( -oo [,) A )  =  ( -oo [,) x )  <->  ( -oo [,) A )  =  ( -oo [,) A ) ) )
1716rspcev 3094 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( -oo [,) A )  =  ( -oo [,) A
) )  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
1814, 17mpan2 671 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
19 eqid 2443 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( -oo [,) x ) )  =  ( x  e.  RR*  |->  ( -oo [,) x ) )
20 ovex 6137 . . . . . . . 8  |-  ( -oo [,) x )  e.  _V
2119, 20elrnmpti 5111 . . . . . . 7  |-  ( ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) )  <->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
2218, 21sylibr 212 . . . . . 6  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
2313, 22sseldi 3375 . . . . 5  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3375 . . . 4  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3375 . . 3  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 466 . 2  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 11327 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 11331 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5585 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6268 . . 3  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  =  (/) )
31 0opn 18539 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2531 . 2  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 164 1  |-  ( -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737    u. cun 3347    C_ wss 3349   (/)c0 3658   ~Pcpw 3881    e. cmpt 4371    X. cxp 4859   ran crn 4862   ` cfv 5439  (class class class)co 6112   +oocpnf 9436   -oocmnf 9437   RR*cxr 9438    < clt 9439    <_ cle 9440   (,)cioo 11321   (,]cioc 11322   [,)cico 11323   topGenctg 14397  ordTopcordt 14458   Topctop 18520   TopBasesctb 18524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-topgen 14403  df-ordt 14460  df-ps 15391  df-tsr 15392  df-top 18525  df-bases 18527  df-topon 18528
This theorem is referenced by: (None)
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